"Minute by Minute" is a great track. I notice these main things about the rhythms in this song:
1. The hi hat playing firmly establishes the triplet pulse in each beat.
2. Once past the intro, the kick always hits on beat 1 (out of 6), and the snare always hits on beat 4 (out of 6). Between points #1 and #2, this firmly establishes a 6/8 meter.
3. The keys player plays a lot of chords that start on the 1st and 3rd beats (out of the 6 beats per measure). This is noticeable at 1:33 and 2:23 in the studio original. It feels like how jazz players play swung 8th notes in a "4/4" time signature (some think of this as dotted 8th followed by a 16th, and it is sometimes written that way, but the actually playing gives about 2/3 of the beat to the first note, and the remaining 1/3 of the beat to the second note).
4. The kick drum joins the keys player in playing on beats 1 and 3 (out of 6) at these places.
5. At other times (not at 1:33 and 2:23) the keys player adds a note played in beat 5 (out of 6). It's as if the keys player is playing an extended triplet in each measure, to offset against the strong emphasis on beats 1 and 4 established by the drummer (see point #2).
6. The drummer also plays a lot on beat 6 out of 6, providing more of the "swung 8th notes" feel described in point #3.
Regarding further discussion of 6/8 shuffles (but not exactly what is played on "Minute by Minute":
Donald Fagen, Walter Becker and Bernard Purdie discuss his "purdie shuffle" as applied to "Home at Last":
https://www.youtube.com/watch?v=_ldtieSEyQM
Bernard Purdie further describing his "purdie shuffle"
https://www.youtube.com/watch?v=T1j1_aeK6WA
Starting at about 2:30 he starts to "'splain" it more, as opposed to showing off.
As a non-drummer, it amazes me that any drummer can make all these rhythms happen at once. To be able to talk about it at the same time, as Purdie does, just blows me away.